Optimization using Calculus

1. Optimization using Calculus Optimization of Functions of Multiple Variables subject to Equality Constraints Optimization Methods: M2L4

2. Objectives • Optimization of functions of multiple variables subjected to equality constraints using • the method of constrained variation, and • the method of Lagrange multipliers. Optimization Methods: M2L4

3. Constrained optimization A function of multiple variables, f(x), is to be optimized subject to one or more equality constraints of many variables. These equality constraints, gj(x), may or may not be linear. The problem statement is as follows: Maximize (or minimize) f(X), subject to gj(X) = 0, j = 1, 2, … , m where Optimization Methods: M2L4

4. Constrained optimization (contd.) • With the condition that ; or else if m > n then the problem becomes an over defined one and there will be no solution. Of the many available methods, the method of constrained variation and the method of using Lagrange multipliers are discussed. Optimization Methods: M2L4

5. Solution by method of Constrained Variation • For the optimization problem defined above, let us consider a specific case with n = 2 and m = 1 before we proceed to find the necessary and sufficient conditions for a general problem using Lagrange multipliers. The problem statement is as follows: Minimize f(x1,x2), subject to g(x1,x2) = 0 • For f(x1,x2) to have a minimum at a point X* = [x1*,x2*], a necessary condition is that the total derivative of f(x1,x2) must be zero at [x1*,x2*]. (1) Optimization Methods: M2L4

6. Method of Constrained Variation (contd.) • Since g(x1*,x2*) = 0 at the minimum point, variations dx1 and dx2 about the point [x1*, x2*] must be admissible variations, i.e. the point lies on the constraint: g(x1* + dx1, x2* + dx2) = 0(2) assuming dx1 and dx2 are small the Taylor series expansion of this gives us (3) Optimization Methods: M2L4

7. Method of Constrained Variation (contd.) or at [x1*,x2*](4) which is the condition that must be satisfied for all admissible variations. Assuming , (4)can be rewritten as (5) Optimization Methods: M2L4

8. Method of Constrained Variation (contd.) (5) indicates that once variation along x1 (dx1) is chosen arbitrarily, the variation along x2 (dx2) is decided automatically to satisfy the condition for the admissible variation. Substituting equation (5)in(1) we have: (6) The equation on the left hand side is called the constrained variation of f. Equation (5)has to be satisfied for all dx1, hence we have (7) This gives us the necessary condition to have [x1*, x2*] as an extreme point (maximum or minimum) Optimization Methods: M2L4

9. Solution by method of Lagrange multipliers Continuing with the same specific case of the optimization problem with n = 2 and m = 1 we define a quantity λ, called the Lagrange multiplier as (8) Using this in (5) (9) And (8) written as (10) Optimization Methods: M2L4

10. Solution by method of Lagrange multipliers…contd. Also, the constraint equation has to be satisfied at the extreme point (11) Hence equations (9) to (11) represent the necessary conditions for the point [x1*, x2*] to be an extreme point. Note that λ could be expressed in terms of as well and has to be non-zero. Thus, these necessary conditions require that at least one of the partial derivatives of g(x1 , x2) be non-zero at an extreme point. Optimization Methods: M2L4

11. Solution by method of Lagrange multipliers…contd. The conditions given by equations (9) to (11) can also be generated by constructing a functions L, known as the Lagrangian function, as (12) Alternatively, treating L as a function of x1,x2 and , the necessary conditions for its extremum are given by (13) Optimization Methods: M2L4

12. Necessary conditions for a general problem For a general problem with n variables and m equality constraints the problem is defined as shown earlier Maximize (or minimize) f(X), subject to gj(X) = 0, j = 1, 2, … , m where In this case the Lagrange function, L, will have one Lagrange multiplier jfor each constraint as (14) Optimization Methods: M2L4

13. Necessary conditions for a general problem…contd. L is now a function of n + m unknowns, , and the necessary conditions for the problem defined above are given by (15) which represent n + m equations in terms of the n + m unknowns, xi and j. The solution to this set of equations gives us (16) and Optimization Methods: M2L4

14. Sufficient conditions for a general problem A sufficient condition for f(X) to have a relative minimum at X* is that each root of the polynomial in Є, defined by the following determinant equation be positive. (17) Optimization Methods: M2L4

15. Sufficient conditions for a general problem…contd. where (18) Similarly, a sufficient condition for f(X) to have a relative maximum at X* is that each root of the polynomial in Є, defined by equation (17) be negative. If equation (17), on solving yields roots, some of which are positive and others negative, then the point X* is neither a maximum nor a minimum. Optimization Methods: M2L4

16. Example Minimize , Subject to or Optimization Methods: M2L4

19. Thank you Optimization Methods: M2L4